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9.5 Review Exercises and Sample Exam

Review Exercises

    Introduction to Sequences and Series

      Find the first 5 terms of the sequence as well as the 30th term.

    1. an=5n3

    2. an=4n+3

    3. an=10n

    4. an=3n

    5. an=(1)n(n2)2

    6. an=(1)n2n1

    7. an=2n+1n

    8. an=(1)n+1(n1)

      Find the first 5 terms of the sequence.

    1. an=nxn2n+1

    2. an=(1)n1xn+2n

    3. an=2nx2n

    4. an=(3x)n1

    5. an=an1+5 where a1=0

    6. an=4an1+1 where a1=2

    7. an=an23an1 where a1=0 and a2=3

    8. an=5an2an1 where a1=1 and a2=0

      Find the indicated partial sum.

    1. 1, 4, 7, 10, 13,…; S5

    2. 3, 1, −1, −3, −5,…; S5

    3. −1, 3, −5, 7, −9,…; S4

    4. an=(1)nn2; S4

    5. an=3(n2)2; S4

    6. an=(15)n2; S4

      Evaluate.

    1. k=16(12k)
    2. k=14(1)k3k2
    3. n=13n+1n
    4. n=175(1)n1
    5. k=48(1k)2
    6. k=22(23)k

    Arithmetic Sequences and Series

      Write the first 5 terms of the arithmetic sequence given its first term and common difference. Find a formula for its general term.

    1. a1=6; d=5

    2. a1=5; d=7

    3. a1=5; d=3

    4. a1=32; d=12

    5. a1=34; d=34

    6. a1=3.6; d=1.2

    7. a1=7; d=0

    8. a1=1; d=1

      Given the terms of an arithmetic sequence, find a formula for the general term.

    1. 10, 20, 30, 40, 50,…

    2. −7, −5, −3, −1, 1,…

    3. −2, −5, −8, −11, −14,…

    4. 13, 0, 13, 23, 1,…

    5. a4=11 and a9=26

    6. a5=5 and a10=15

    7. a6=6 and a24=15

    8. a3=1.4 and a7=1

      Calculate the indicated sum given the formula for the general term of an arithmetic sequence.

    1. an=4n3; S60

    2. an=2n+9; S35

    3. an=15n12; S15

    4. an=n+14; S20

    5. an=1.8n4.2; S45

    6. an=6.5n+3; S35

      Evaluate.

    1. n=122(7n5)
    2. n=1100(14n)
    3. n=135(23n)
    4. n=130(14n+1)
    5. n=140(2.3n1.1)
    6. n=1300n
    7. Find the sum of the first 175 positive odd integers.

    8. Find the sum of the first 175 positive even integers.

    9. Find all arithmetic means between a1=23 and a5=23

    10. Find all arithmetic means between a3=7 and a7=13.

    11. A 5-year salary contract offers $58,200 for the first year with a $4,200 increase each additional year. Determine the total salary obligation over the 5-year period.

    12. The first row of seating in a theater consists of 10 seats. Each successive row consists of four more seats than the previous row. If there are 14 rows, how many total seats are there in the theater?

    Geometric Sequences and Series

      Write the first 5 terms of the geometric sequence given its first term and common ratio. Find a formula for its general term.

    1. a1=5; r=2

    2. a1=3; r=2

    3. a1=1; r=32

    4. a1=4; r=13

    5. a1=1.2; r=0.2

    6. a1=5.4; r=0.1

      Given the terms of a geometric sequence, find a formula for the general term.

    1. 4, 40, 400,…

    2. −6, −30, −150,…

    3. 6, 92, 278,…

    4. 1, 35, 925,…

    5. a4=4 and a9=128

    6. a2=1 and a5=64

    7. a2=52 and a5=62516

    8. a3=50 and a6=6,250

    9. Find all geometric means between a1=1 and a4=64.

    10. Find all geometric means between a3=6 and a6=162.

      Calculate the indicated sum given the formula for the general term of a geometric sequence.

    1. an=3(4)n1; S6

    2. an=5(3)n1; S10

    3. an=32(2)n; S14

    4. an=15(3)n+1; S12

    5. an=8(12)n+2; S8

    6. an=18(2)n+2; S10

      Evaluate.

    1. n=1103(4)n
    2. n=1935(2)n1
    3. n=13(23)n
    4. n=112(45)n+1
    5. n=112(32)n
    6. n=132(12)n
    7. After the first year of operation, the value of a company van was reported to be $40,000. Because of depreciation, after the second year of operation the van was reported to have a value of $32,000 and then $25,600 after the third year of operation. Write a formula that gives the value of the van after the nth year of operation. Use it to determine the value of the van after 10 years of operation.

    8. The number of cells in a culture of bacteria doubles every 6 hours. If 250 cells are initially present, write a sequence that shows the number of cells present after every 6-hour period for one day. Write a formula that gives the number of cells after the nth 6-hour period.

    9. A ball bounces back to one-half of the height that it fell from. If dropped from 32 feet, approximate the total distance the ball travels.

    10. A structured settlement yields an amount in dollars each year n according to the formula pn=12,500(0.75)n1. What is the total value of a 10-year settlement?

      Classify the sequence as arithmetic, geometric, or neither.

    1. 4, 9, 14,…

    2. 6, 18, 54,…

    3. −1, 12, 0,…

    4. 10, 30, 60,…

    5. 0, 1, 8,…

    6. −1, 23, 49,…

      Evaluate.

    1. n=14n2
    2. n=14n3
    3. n=132(4n+5)
    4. n=12(15)n1
    5. n=1813(3)n
    6. n=146(14n12)
    7. n=122(3n)
    8. n=1312n
    9. n=1283
    10. n=1303(1)n1
    11. n=1313(1)n1

    Binomial Theorem

      Evaluate.

    1. 8!

    2. 11!

    3. 10!2!6!

    4. 9!3!8!

    5. (n+3)!n!

    6. (n2)!(n+1)!

      Calculate the indicated binomial coefficient.

    1. (74)
    2. (83)
    3. (105)
    4. (1110)
    5. (120)
    6. (n+1n1)
    7. (nn2)

      Expand using the binomial theorem.

    1. (x+7)3

    2. (x9)3

    3. (2y3)4

    4. (y+4)4

    5. (x+2y)5

    6. (3xy)5

    7. (uv)6

    8. (u+v)6

    9. (5x2+2y2)4

    10. (x32y2)4

Answers

  1. 2, 7, 12, 17, 22; a30=147

  2. −10, −20, −30, −40, −50; a30=300

  3. −1, 0, −1, 4, −9; a30=784

  4. 3, 52, 73, 94, 115; a30=6130

  5. x3,2x25,3x37,4x49,5x511

  6. 2x2,4x4,8x6,16x8,32x10

  7. 0, 5, 10, 15, 20

  8. 0, −3, 9, −30, 99

  9. 35

  10. −5

  11. −18

  12. −36

  13. 296

  14. 135

  1. 6, 11, 16, 21, 26; an=5n+1

  2. 5, 2, −1, −4, −7; an=83n

  3. 34, 32, 94, −3, 154; an=34n

  4. 7, 7, 7, 7, 7; an=7

  5. an=10n

  6. an=13n

  7. an=3n1

  8. an=12n+3

  9. 7,140

  10. 332

  11. 1,674

  12. 1,661

  13. 420

  14. 1,842

  15. 30,625

  16. 13, 0, 13

  17. $333,000

  1. 5, 10, 20, 40, 80; an=5(2)n1

  2. 1, 32, 94, 278, 8116; an=(32)n1

  3. 1.2, 0.24, 0.048, 0.0096, 0.00192; an=1.2(0.2)n1

  4. an=4(10)n1

  5. an=6(34)n1

  6. a1=12(2)n1

  7. an=(52)n1

  8. 4, −16

  9. 4,095

  10. 16,383

  11. 255128

  12. 2,516,580

  13. −6

  14. No sum

  15. vn=40,000(0.8)n1; v10=$5,368.71

  16. 96 feet

  17. Arithmetic; d=5

  18. Arithmetic; d=12

  19. Neither

  20. 30

  21. −1,952

  22. 1,640

  23. −187

  24. 84

  25. 3

  1. 39,916,800

  2. 54

  3. 1n(n+1)(n1)

  4. 56

  5. 11

  6. n(n+1)2

  7. x3+21x2+147x+343

  8. 16y496y3+216y2216y+81

  9. x5+10x4y+40x3y2+80x2y3+80xy4+32y5

  10. u66u5v+15u4v220u3v3+15u2v46uv5+v6
  11. 625x8+1,000x6y2+600x4y4+160x2y6+16y8

Sample Exam

      Find the first 5 terms of the sequence.

    1. an=6n15

    2. an=5(4)n2

    3. an=n12n1

    4. an=(1)n1x2n

      Find the indicated partial sum.

    1. an=(n1)n2; S4

    2. k=15(1)k2k2

      Classify the sequence as arithmetic, geometric, or neither.

    1. −1, 32, −2,…

    2. 1, −6, 36,…

    3. 38, 34, 32,…

    4. 12, 14, 29,…

      Given the terms of an arithmetic sequence, find a formula for the general term.

    1. 10, 5, 0, −5, −10,…

    2. a4=12 and a9=2

      Given the terms of a geometric sequence, find a formula for the general term.

    1. 18, 12, −2, −8, −32,…

    2. a3=1 and a8=32

      Calculate the indicated sum.

    1. an=5n; S44

    2. an=(2)n+2; S12

    3. n=14(12)n1
    4. n=1100(2n32)

      Evaluate.

    1. 14!10!6!

    2. (97)
    3. Determine the sum of the first 48 positive odd integers.

    4. The first row of seating in a theater consists of 14 seats. Each successive row consists of two more seats than the previous row. If there are 22 rows, how many total seats are there in the theater?

    5. A ball bounces back to one-third of the height that it fell from. If dropped from 27 feet, approximate the total distance the ball travels.

      Expand using the binomial theorem.

    1. (x5y)4

    2. (3a+b2)5

Answers

  1. −9, −3, 3, 9, 15

  2. 0, 13, 25, 37, 49

  3. 70

  4. Arithmetic

  5. Geometric

  6. an=155n

  7. an=18(4)n1

  8. −770

  9. 83

  10. 1,00130

  11. 2,304

  12. 54 feet

  13. 243a5+405a4b2+270a3b4+90a2b6+15ab8+b10